After you run a search in Lexis or Westlaw, you can create an alert to email you new results. Just click on the bell icon (In Lexis, the bell is in the upper left, on Westlaw it is in the upper right).
To view and edit alerts,
Lexis Learn has a tutorial on setting up alerts.
When you run a search in Bloomberg Law, click on the Create Alert link in the upper left. See the Bloomberg Law Workflow Tools video
You can also subscribe to various newsletters to get new content sent to your email. Click on News & Analysis>Bloomberg Law News, and select a topic of interest. Then click on Subscribe to Newsletter in the upper right.